As my family has all gone out shopping, I am left alone to ponder the mysteries of life. My family are Yahtzee fanatics, and I find myself struggling to calculate probabilities for this game. For basic probability, I know the formula is
Chosen Outcome
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# of Possible Outcomes
For example, on a standard d6, there is a 1 out of 6 chance that any number will come up. If I have a 2,3,4,5 and am rolling for a large straight, I should have a 2/6 or 1/3 chance to get it in one roll (since either a 1 or a 6 will work).
IIRC, the odds for multiple dice with separate outcomes for each are calculated by multiplying the individual odds; i.e., the odds for rolling a Yahtzee (all 5 d6 on same number) in a single roll would be 1/6^5, or 1/7776. However, it's been quite some time since I've studied probability, so I don't remember if that's correct.
My dilemma comes when you factor in multiple dice and multiple rolls for a single outcome. Rolling 6 d6 does not guarantee a roll of any particular number. Likewise, rolling 1 d6 six times does not guarantee a roll of any particular number. Obviously, simple experimentation shows there is something more to this than a simple formula such as:
Chosen Outcomes * Rolls
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Possible Outcomes
What, then is the formula? Using the large straight example, given a 2,3,4,5 and 2 rolls of a single die, what is the probability that I will roll a large straight?
I impatiently await your elucid answers to these vital and perplexing questions about life's meaning (at least while my in-laws are here, so I can finally beat them
)!